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A002201 Superior highly composite numbers: positive integers n for which there is an e > 0 such that d(n)/n^e >= d(k)/k^e for all k > 1, where the function d(n) counts the divisors of n (A000005).
(Formerly M1591 N0620)
27

%I M1591 N0620

%S 2,6,12,60,120,360,2520,5040,55440,720720,1441440,4324320,21621600,

%T 367567200,6983776800,13967553600,321253732800,2248776129600,

%U 65214507758400,195643523275200,6064949221531200

%N Superior highly composite numbers: positive integers n for which there is an e > 0 such that d(n)/n^e >= d(k)/k^e for all k > 1, where the function d(n) counts the divisors of n (A000005).

%C For fixed e > 0, d(n)/n^e is bounded and reaches its maximum at one or more points.

%C This is an infinite subset of A002182.

%C The first 15 numbers in this sequence agree with those in A004490 (colossally abundant numbers). - _David Terr_, Sep 29 2004

%D J. L. Nicolas, On highly composite numbers, pp. 215-244 in Ramanujan Revisited, Editors G. E. Andrews et al., Academic Press 1988.

%D S. Ramanujan, Highly composite numbers, Proc. London Math. Soc., 14 (1915), 347-407. Reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, pp. 78-129. See esp. pp. 87, 115.

%D S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.

%D S. Ramanujan, Highly Composite Numbers: Section IV, in 1) Collected Papers of Srinivasa Ramanujan, pp. 111-8, Ed. G. H. Hardy et al., AMS Chelsea 2000. 2) Ramanujan's Papers, pp. 143-150, Ed. B. J. Venkatachala et al., Prism Books Bangalore 2000.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002201/b002201.txt">Table of n, a(n) for n = 1..150</a>

%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram15.html">Highly composite numbers</a>, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper15/page35.htm">IV: Superior Highly Composite Numbers</a>

%H S. Ratering, <a href="http://www.jstor.org/stable/2690653">An interesting subset of the highly composite numbers</a>, Math. Mag., 64 (1991), 343-346.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SuperiorHighlyCompositeNumber.html">Superior Highly Composite Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ColossallyAbundantNumber.html">Colossally Abundant Number</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Superior_highly_composite_number">Superior highly composite number</a>

%e For n=2, 6 and 12 we may take e in the intervals (log(2)/log(3), 1], (log(3/2)/log(2), log(2)/log(3)] and (log(2)/log(5), log(3/2)/log(2)], respectively.

%e Can the intervals in the previous line can be extended to include the left endpoints? - _Ant King_, May 02 2005

%Y Cf. A000705, A004490, A000005.

%Y Cf. A002182, A072938, A106037, A094348, A003418, A002110.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_

%E Better definition from _T. D. Noe_, Nov 05 2002

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Last modified March 25 07:12 EDT 2019. Contains 321468 sequences. (Running on oeis4.)